Advertisement

A Hybrid Algorithm for the Simple Cell Mapping Method in Multi-objective Optimization

  • Yousef Naranjani
  • Carlos Hernández
  • Fu-Rui Xiong
  • Oliver Schütze
  • Jian-Qiao Sun
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 227)

Abstract

This paper presents a hybrid gradient free-gradient (GFG) algorithm for the simple cell mapping (SCM) method for multi-objective optimization problems (MOPs). The SCM method is briefly reviewed in the context of the multi-objective optimization problems (MOPs). We present a mixed application of gradient free directed search and gradient search algorithms for the SCM method and discuss its potentials for higher dimensional MOPs. We present several numerical exmaples to demonstrate the effectiveness of the proposed hybrid algorithm. The examples include two simple geometric MOPs, an example with five design parameters, and a proportional-integral-derivative (PID) control design for a second order linear system.

Keywords

Simple cell mapping Multi-objective optimization Feedback control 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lara, A., Alvarado, S., Salomon, S., Avigad, G., Coello Coello, C.A., Schütze, O.: The gradient free directed search method as local search within multi-objective evolutionary algorithms. In: Schütze, O., Coello Coello, C.A., Tantar, A.-A., Tantar, E., Bouvry, P., Del Moral, P., Legrand, P. (eds.) EVOLVE - A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation II. AISC, vol. 175, pp. 153–168. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Coello Coello, C.A., Lamont, G., Van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-Objective Problems. Springer, New York (2007)zbMATHGoogle Scholar
  3. 3.
    Schütze, O., Vasile, M., Coello Coello, C.A.: Computing the set of epsilon-efficient solutions in multiobjective space mission design. Journal of Aerospace Computing, Information, and Communication 8(3), 53–70 (2011)CrossRefGoogle Scholar
  4. 4.
    Dellnitz, M., Schütze, O., Hestermeyer, T.: Covering Pareto sets by multilevel subdivision techniques. Journal of Optimization Theory and Applications 124, 113–155 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Jahn, J.: Multiobjective search algorithm with subdivision technique. Computational Optimization and Applications 35(2), 161–175 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Schütze, O., Vasile, M., Junge, O., Dellnitz, M., Izzo, D.: Designing optimal low thrust gravity assist trajectories using space pruning and a multi-objective approach. Engineering Optimization 41(2), 155–181 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hsu, C.S.: Cell-to-Cell Mapping, A Method of Global Analysis for Nonlinear Systems. Springer, New York (1987)zbMATHCrossRefGoogle Scholar
  8. 8.
    Hsu, C.S.: A theory of cell-to-cell mapping dynamical systems. Journal of Applied Mechanics 47, 931–939 (1980)zbMATHCrossRefGoogle Scholar
  9. 9.
    Hsu, C.S.: A discrete method of optimal control based upon the cell state space concept. Journal of Optimization Theory and Applications 46(4), 547–569 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bursal, F.H., Hsu, C.S.: Application of a cell-mapping method to optimal control problems. International Journal of Control 49(5), 1505–1522 (1989)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Crespo, L.G., Sun, J.Q.: Stochastic optimal control of nonlinear dynamic systems via bellman’s principle and cell mapping. Automatica 39(12), 2109–2114 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Flashner, H., Burns, T.F.: Spacecraft momentum unloading: the cell mapping approach. Journal of Guidance, Control and Dynamics 13, 89–98 (1990)CrossRefGoogle Scholar
  13. 13.
    Zhu, W.H., Leu, M.C.: Planning optimal robot trajectories by cell mapping. In: Proceedings of Conference on Robotics and Automation, pp. 1730–1735 (1990)Google Scholar
  14. 14.
    Wang, F.Y., Lever, P.J.A.: A cell mapping method for general optimum trajectory planning of multiple robotic arms. Robotics and Autonomous Systems 12, 15–27 (1994)CrossRefGoogle Scholar
  15. 15.
    Yen, J.Y.: Computer disk file track accessing controller design based upon cell to cell mapping. In: Proceedings of the American Control Conference (1992)Google Scholar
  16. 16.
    Crespo, L.G., Sun, J.Q.: Solution of fixed final state optimal control problems via simple cell mapping. Nonlinear Dynamics 23, 391–403 (2000)zbMATHCrossRefGoogle Scholar
  17. 17.
    Crespo, L.G., Sun, J.Q.: Optimal control of target tracking via simple cell mapping. Journal of Guidance and Control 24, 1029–1031 (2000)CrossRefGoogle Scholar
  18. 18.
    Crespo, L.G., Sun, J.Q.: Fixed final time optimal control via simple cell mapping. Nonlinear Dynamics 31, 119–131 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Pareto, V.: Manual of Political Economy. The MacMillan Press, London (1971); original edition in French (1927)Google Scholar
  20. 20.
    Hillermeier, C.: Nonlinear Multiobjective Optimization - A Generalized Homotopy Approach. Birkhäuser, Berlin (2001)zbMATHCrossRefGoogle Scholar
  21. 21.
    Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Mathematical Methods of Operations Research 51(3), 479–494 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Liu, G.P., Yang, J.B., Whidborne, J.F.: Multiobjective Optimisation and Control. Research Studies Press, Baldock (2002)Google Scholar
  23. 23.
    Schütze, O., Lara, A., Coello Coello, C.A.: The directed search method for unconstrained multi-objective optimization problems. In: Proceedings of the EVOLVE – A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation (2011)Google Scholar
  24. 24.
    Lara, A., Alvarado, S., Salomon, S., Avigad, G., Coello Coello, C.A., Schütze, O.: The gradient free directed search method as local search within multi-objective evolutionary algorithms. In: Schütze, O., Coello Coello, C.A., Tantar, A.-A., Tantar, E., Bouvry, P., Del Moral, P., Legrand, P. (eds.) EVOLVE - A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation II. AISC, vol. 175, pp. 153–168. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  25. 25.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer (2006)Google Scholar
  26. 26.
    Das, I., Dennis, J.: Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM Journal of Optimization 8, 631–657 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Schäffler, S., Schultz, R., Weinzierl, K.: A stochastic method for the solution of unconstrained vector opimization problems. Journal of Optimization, Theory and Application 114(1), 209–222 (2002)zbMATHCrossRefGoogle Scholar
  28. 28.
    Liu, G.P., Daley, S.: Optimal-tuning PID controller design in the frequency domain with application to a rotary hydraulic system. Control Engineering Practice 7(7), 821–830 (1999)CrossRefGoogle Scholar
  29. 29.
    Liu, G.P., Daley, S.: Optimal-tuning nonlinear PID control of hydraulic systems. Control Engineering Practice 8(9), 1045–1053 (2000)CrossRefGoogle Scholar
  30. 30.
    Panda, S.: Multi-objective PID controller tuning for a FACTS-based damping stabilizer using non-dominated sorting genetic algorithm-II. International Journal of Electrical Power & Energy Systems 33(7), 1296–1308 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Yousef Naranjani
    • 1
  • Carlos Hernández
    • 3
  • Fu-Rui Xiong
    • 2
  • Oliver Schütze
    • 3
  • Jian-Qiao Sun
    • 1
  1. 1.School of EngineeringUniversity of California at MercedMercedUSA
  2. 2.Department of MechanicsTianjin UniversityTianjinChina
  3. 3.Depto de ComputacionCINVESTAV-IPNMexico CityMexico

Personalised recommendations